Question: Bernardo and Ogechi were asked to find an explicit formula for the sequence $1\,,\,8\,,\,64\,,\,512,...$, where the first term should be $h(1)$. Bernardo said the formula is $h(n)=1\cdot8^{{n}}$, and Ogechi said the formula is $h(n)=8\cdot1^{{n}}$. Which one of them is right? Choose 1 answer: Choose 1 answer: (Choice A) A Only Bernardo (Choice B) B Only Ogechi (Choice C) C Both Bernardo and Ogechi (Choice D) D Neither Bernardo nor Ogechi
Answer: In a geometric sequence, the ratio between successive terms is constant. This means that we can move from any term to the next one by multiplying by a constant value. Let's calculate this ratio over the first few terms: $\dfrac{512}{64}=\dfrac{64}{8}=\dfrac{8}{1}={8}$ We see that the constant ratio between successive terms is ${8}$. In other words, we can find any term by starting with the first term and multiplying by ${8}$ repeatedly until we get to the desired term. Let's look at the first few terms expressed as products: $n$ $1$ $2$ $3$ $4$ $h(n)$ ${1}\cdot\!{8}^{0}$ ${1}\cdot\!{8}^{1}$ ${1}\cdot\!{8}^{2}$ ${1}\cdot\!{8}^{3}$ We can see that every term is the product of the first term, ${1}$, and a power of the constant ratio, ${8}$. Note that this power is always one less than the term number $n$. This is because the first term is the product of itself and plainly $1$, which is like taking the constant ratio to the zeroth power. Thus, we arrive at the following explicit formula (Note that ${1}$ is the first term and ${8}$ is the constant ratio): $h(n)={1}\cdot{8}^{{\,n-1}}$ We can now see that $h(n)=1\cdot8^{{\,n}}$ is not a correct formula, because the constant ratio is multiplied one extra time for each term. For instance, according to this formula, the value of the first term would be: $h(1)=1\cdot8^{{\,1}} = 8$. However, according to our table of values, $h(1)=1$. So Bernardo is definitely wrong. What about Ogechi? We can see that $h(n)=8\cdot1^{{\,n}}$ is also not a correct formula, because the constant ratio according to this formula is $1$, while the actual constant ratio is $8$. Hence, Ogechi is also wrong. Neither Bernardo nor Ogechi got a correct explicit formula.